Proposition 5.4 (cf. [1]}) 

Let \(G_{\bf s} = \mathbb {Z}_{s_1} \times \cdots \times \mathbb {Z}_{s_{m}}\) ,
\(G_{\bf t} = \mathbb {Z}_{t_1} \times \cdots \times \mathbb {Z}_{t_{n}}\) , and
\(G = G_{\bf s} \times G_{\bf t}\) .
Suppose that
\(\psi \partial \phi _{\bf s}\in Z^2( G_{\bf s}, \langle \zeta _{k_{1}} \rangle )\)  and
\(\rho \partial \phi _{\bf t}\in Z^2( G_{\bf t},\langle \zeta _{k_{2}} \rangle )\) 
are orthogonal.
Let \(k = \mathrm {lcm}(k_{1},k_{2})\) .
Define \(\varphi \in Z^2(G, \langle \zeta _{k} \rangle )\)  by
\(\varphi (g_{s}g_{t},h_{s}h_{t}) =\psi (g_{s},h_{s})\rho (g_{t},h_{t})\) ,
and define a map \(\phi \)  on \(G\) 
by \(\phi (g_{s}g_{t}) =\phi _{\bf s}(g_{s}) \phi _{\bf t}(g_{t})\) .
Then \(\varphi \partial \phi \in Z^2(G,\langle \zeta _{k} \rangle )\) 
is orthogonal, with cocyclic matrix
\([\psi \partial \phi _{\bf s}]\otimes [\rho \partial \phi _{\bf t}]\) .

